Moving beyond linear summation to infer interaction order from neural and biological dynamics.

We introduce the three-body recurrent neural network (TBRNN), a recurrent model that explicitly incorporates quadratic, three-body interactions. We show that TBRNNs are universal approximators and extend low-rank recurrent neural network (RNN) theory to derive a corresponding low-rank TBRNN framework, allowing model rank to help discriminate between pairwise and higher-order dynamics. On canonical neuroscience tasks, TBRNNs exhibit solution geometries distinct from standard RNNs, indicating that higher-order interactions reshape accessible dynamical regimes rather than merely re-parameterizing pairwise models. Building on these results, we develop a practical model-comparison procedure that infers interaction order directly from observed trajectories. Applied to synthetic systems, a gene regulatory model, and neural recordings, the framework distinguishes pairwise, three-body, and mixed interaction structure. Our results broaden the space of interpretable dynamical models in neuroscience and provide a general approach for probing higher-order interactions in biological networks.